Let $y=f(x)$ be a continuous function.
We can see as $∆x →0$ ,we have $∆y→0$ (by definition of continuous function)
My question is does as $∆y→0$ can $∆x→0$? Please explain it .Thanks.
doubts regarding continuity
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calculus
real-analysis
limits
continuity
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0The property holds if the function is invertible – 2017-01-27
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0One possibility is ∆x→0 .Do ∆x tend to any other value since the curve may doubles back – 2017-01-27
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0So if the function is both one one and onto this property holds? Right – 2017-01-27
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0@SathasivamK, that works OK for functions of one dimension, but watch out for higher dimensional cases. The function which maps $[0,1)$ to the circle $\{(a,b) \in \mathbb R^2 \mid a^2 + b^2 = 1\}$ by $x \mapsto (\cos(2\pi x),\sin(2 \pi x))$ is continuous, one-to-one and onto, but you do not have $\Delta y \to 0 \implies \Delta x \to 0$. – 2017-01-27
1 Answers
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Hint: Think about a constant function, $\Delta y$ is always $0$ and $\Delta x$ can be any number.